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Under the assumption that the complexification $X_{\\mathbb C}$ is $L_{\\mathbb C}$-spherical, we prove an explicit correspondence between spectral analysis on the standard locally homogeneous space $X_{\\Gamma}=\\Gamma\\backslash X$ and on $\\Gamma\\backslash L$ via branching laws for the restriction to $L$ of irreducible representations of $G$. 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